Question

multistep function inverses (level 1)
score:0/3 penalty:none
question
for the function $f(x)=\frac{x^\frac{1}{3}}{5}$, find $f^{-1}(x)$.
answer
$\circ$ $f^{-1}(x)=\frac{x^3}{5}$
$\circ$ $f^{-1}(x)=(5x)^3$
$\circ$ $f^{-1}(x)=5x^3$
$\circ$ $f^{-1}(x)=\left(\frac{x}{5}\
ight)^3$

Explanation:

Step1: Set $y = f(x)$

$y = \frac{x^{\frac{1}{3}}}{5}$

Step2: Swap $x$ and $y$

$x = \frac{y^{\frac{1}{3}}}{5}$

Step3: Solve for $y$, multiply by 5

$5x = y^{\frac{1}{3}}$

Step4: Cube both sides

$(5x)^3 = y$

Step5: Replace $y$ with $f^{-1}(x)$

$f^{-1}(x) = (5x)^3$

Answer:

$f^{-1}(x) = (5x)^3$ (corresponding to the option $\boldsymbol{f^{-1}(x) = (5x)^3}$)